Quadrature radio frequency (RF) signal transmission and reception exploits the concept that a single carrier wave may carry two independent data streams, provided that the data streams modulate the carrier in quadrature. For example, one of the data streams may modulate the carrier as a sine function and the other data stream may modulate the carrier as a cosine function. In any case, the two modulation streams must have a phase offset of 90° with respect to each other. The quadrature data streams are typically referred to as I data and Q data, representing an in-phase data stream and a data stream that is in quadrature with the in-phase data stream, that is, at a phase angle of 90° thereto.
A quadrature RF receiver splits the received signal into two paths, referred to herein as an “I arm” and a “Q arm.” The I signal is down-converted by mixing the received signal with an I-arm local oscillator (LO) carrier signal. The Q signal is likewise down-converted by mixing the received signal with a Q-arm LO carrier. Ideally, the Q-arm LO signal is of precisely the same magnitude as the I-arm LO signal and is exactly 90° out of phase with the I-arm LO signal. In actual practice, the I and Q arm LO signals often vary somewhat in magnitude and drift in phase away from perfect quadrature, creating a problem referred to as I/Q mismatch. Other factors such as non-ideal low-pass filter characteristics may also contribute to I/Q mismatch.
I/Q mismatch results in extraneous LO energy components in the image spectrum and the subsequent down-conversion of image spectrum interference. Said differently, I/Q mismatch results in decreased image rejection performance and lower SNRs.
In the presence of gain and phase mismatch, I and Q LO carriers may be mathematically modeled as:CI(t)=cos(ωt+θ);CQ(t)=−β sin(ωt+θ+φ)where φ represents phase mismatch between the two LO carriers and where β represents gain mismatch between the two LO carriers.
The resulting received, down-converted I and Q signals may be represented as:IR(t)=I(t)QR(t)=β(Q(t)cos φ−I(t)sin φ)It is noted that the I and Q signals are not independent in the presence of I/Q mismatch as they are when in ideal quadrature with each other. Rather, the Q signal now includes an I signal component.
The gain imbalance β is estimated as:
  β  =                              variance          (                      Q            R            2                    )                          variance          (                      I            R            2                    )                      =                            σ                      Q            R            2                                    σ                      I            R            2                              
The phase imbalance φ is estimated as:
  ϕ  =            sin              -        1              ⁡          (                                    E            ⁡                          [                                                I                  R                                ·                                  Q                  R                                            ]                                -                                    E              ⁡                              [                                  I                  R                                ]                                      ·                          E              ⁡                              [                                  Q                  R                                ]                                                                          σ                          I              R                                ⁢                      σ                          Q              R                                          )      